C₂×#758216 #72337863

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂×#758216

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

61705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

(8,9)(11,12,15,17)(13,16,14,18)(20,22)(24,26);(11,13)(12,16)(14,15)(17,18)(19,20,23,22)(21,24,25,26);(4,6)(7,8,10,9)(12,17)(13,14);(4,6)(7,8,10,9)(11,14,15,13)(12,18,17,16)(19,21)(20,24)(22,26)(23,25);(1,2);(3,4,5,6)(7,9,10,8)(11,12)(13,18)(14,16)(15,17)(19,22,23,20)(21,24,25,26)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72210448, C₂×#72232334
maximal normal subgroup of	C₂×#72210448, C₂×#72232334

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((8,9)(11,12,15,17)(13,16,14,18)(20,22)(24,26), (11,13)(12,16)(14,15)(17,18)(19,20,23,22)(21,24,25,26), (4,6)(7,8,10,9)(12,17)(13,14), (4,6)(7,8,10,9)(11,14,15,13)(12,18,17,16)(19,21)(20,24)(22,26)(23,25), (1,2), (3,4,5,6)(7,9,10,8)(11,12)(13,18)(14,16)(15,17)(19,22,23,20)(21,24,25,26));

C2×#758216 #72337863